How to render an object without its shadow?
One option is turning off the Shadow property per object (Properties > Object > Cycles Settings):



As of Blender 2.8x the Cycles Settings panel has been renamed to Visibility and the Shadow property is now part of the new Ray Visibility subpanel. For Eevee you can only turn off the Shadow property per light for the time being (Properties > Object Data > Light):





Another option for Cycles is using the Light Path node. Plug Is Shadow Ray into the Factor socket of a Mix Shader and use a Transparent BSDF for the second Shader input:



Result